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Journal of Mathematical Sciences

, Volume 234, Issue 1, pp 1–13 | Cite as

Separating transformation and extremal problems on nonoverlapping simply connected domains

  • Aleksandr K. Bakhtin
Article
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Abstract

We consider the well-known problem of maximum of the functional
$$ {I}_n\left(\upgamma \right)={r}^{\upgamma}\left({B}_0.0\right)\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), $$

where B0, …, Bn are pairwise disjoint domains in \( \overline{\mathrm{\mathbb{C}}} \), a0 = 0, |ak| = 1, \( k=\overline{1,n} \), are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ.

Keywords

Inner radius of a domain nonoverlapping domains radial systems of points control functional separating transformation quadratic differential Green function 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics of the NAS of UkraineKievUkraine

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